The dynamic analysis of prestressed, bending-torsion coupled beams is revisited. The axially loaded beam is assumed to be slender, isotropic, homogeneous, and linearly elastic, exhibiting coupled flexural-torsional displacement caused by the end moment. Based on the Euler-Bernoulli bending and St. Venant torsion beam theories, the vibration and stability of such beams are explored. Using the closed-form solutions of the uncoupled portions of the governing equations as the basis functions of approximation space, the dynamic, frequency-dependent, interpolation functions are developed, which are then used in conjunction with the weighted residual method to develop the Dynamic Finite Element (DFE) of the system. Having implemented the DFE in a MATLAB-based code, the resulting nonlinear eigenvalue problem is then solved to determine the coupled natural frequencies of illustrative beam examples, subjected to various boundary and load conditions. The proposed method is validated against limited available experimental and analytical data, those obtained from an in-house conventional Finite Element Method (FEM) code and FEM-based commercial software (ANSYS). In comparison with FEM, the DFE exhibits higher convergence rates and in the absence of end moment it produces exact results. Buckling analysis is also carried out to determine the critical end moment and compressive force for various load combinations.
The free vibration of beams, subjected to a constant axial load and end moment and various boundary conditions, is examined. Based on the Euler-Bernoulli bending and St. Venant torsion beam theories, the differential equations governing coupled flexural-torsional vibrations and stability of a uniform, slender, isotropic, homogeneous, and linearly elastic beam, undergoing linear harmonic vibration, are first reviewed. The existing formulations are then briefly discussed and a conventional finite element method (FEM) is developed. Exploiting the MATLAB-based code, the resulting linear Eigenvalue problem is then solved to determine the Eigensolutions (i.e., natural frequencies and modes) of illustrative examples, exhibiting geometric bending-torsion coupling. Various classical boundary conditions are considered and the FEM frequency results are validated against those obtained from a commercial software (ANSYS) and the data available in the literature. Tensile axial force is found to increase natural frequencies, indicating beam stiffening. However, when a force and an end moment are acting in combination, the moment reduces the stiffness of the beam and the stiffness of the beam is found to be more sensitive to the changes in the magnitude of the axial force compared to the moment. A buckling analysis of the beam is also carried out to determine the critical buckling end moment and axial compressive force.
Free vibration analysis of beams with single delamination undergoing bending-torsion coupling is made, using traditional finite element technique. The Galerkin weighted residual method is applied to convert the coupled differential equations of motion into to a discrete problem, where, in addition to the conventional mass and stiffness matrices, a delamination stiffness matrix, representing the extra stiffening effects at the delamination tips, is introduced. The linear eigenvalue problem resulting from the discretization along the length of the beam is solved to determine the frequencies and modes of free vibration. Both “free mode” and “constrained mode” delamination models are considered in formulation, and it is shown that the continuity (both kinematic and force) conditions at the beam span-wise locations corresponding to the extremities of the delaminated region, in particular, play a great role in “free mode” model formulation. Current trends in the literature are examined, and insight into different types of modeling techniques and constraint types are introduced. In addition, the data previously available in the literature and those obtained from a finite element-based commercial software are utilized to validate the presented modeling scheme and to verify the correctness of natural frequencies of the systems analyzed here. The paper ends with general discussions and conclusions on the presented theories and modeling approaches.
The free vibration of beams, subjected to a constant axial load and end moment and various boundary conditions, is examined. Based on the Euler-Bernoulli bending and St. Venant torsion beam theories, the differential equations governing coupled flexural-torsional vibrations and stability of a uniform, slender, isotropic, homogeneous, and linearly elastic beam, undergoing linear harmonic vibration, are first reviewed. The existing formulations are then briefly discussed and a conventional finite element method (FEM) is developed. Exploiting the MAT LAB-based code, the resulting linear Eigenvalue problem is then solved to determine the Eigensolutions (i.e., natural frequencies and modes) of illustrative examples, exhibiting geometric bending-torsion coupling. Various classical boundary conditions are considered and the FEM frequency results are validated against those obtained from a commercial software (ANSYS) and the data available in the literature. Tensile axial force is found to increase natural frequencies, indicating beam stiffening. However, when a force and an end moment are acting in combination, the moment reduces the stiffness of the beam and the stiffness of the beam is found to be more sensitive to the changes in the magnitude of the axial force compared to the moment. A buckling analysis of the beam is also carried out to determine the critical buckling end moment and axial compressive force.
This work presents the bending–torsion coupled free vibration analysis of prestressed, layered composite beams subjected to axial force and end moment using the traditional finite element method (FEM) and dynamic finite element (DFE) techniques. Current trends in the literature, in terms of different types of modeling techniques and constraints, were briefly examined. The Galerkin-type weighted residual method was applied to convert the coupled differential equations of motion into a discrete problem using a polynomial interpolation function in the finite element method. In the dynamic finite element method, trigonometric shape functions were implemented to describe the equations in terms of nodal displacements. The eigenvalue problem resulting from the discretization along the length of the beam was solved in order to determine the system’s natural frequencies and modes. The results, showing the effects of axial load, end moment, and combined loading on natural frequencies, are discussed and are followed by some concluding remarks. <br><div><br></div>
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.